Keywords: The mean value theorem, Iteration, Sturm function, Complex differential equation, Oscillation. 1 Introduction and Preliminary Notes. It is common that ...
Applied Mathematical Sciences, Vol. 8, 2014, no. 37, 1817 - 1827 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4279
The Mean Value Theorem of Line Complex Integral and Sturm Function Jelena Vujakovi´ c Faculty of Sciences and Mathematics, University of Priˇstina Lole Ribara 29, 38 220 Kosovska Mitrovica, Serbia c 2014 Jelena Vujakovi´c. This is an open access article distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract For linear complex differential equation of second order w (z) + a (z) = 0, where a (z) is analytical function, two solutions w1 ,w2 are obtained by iterations (according to results from [4] where are present successive line integrals on unclosed arc L of integration path. If L would be contour, then due to analytics and Cauchy theorem every first integral in order in these successive integrals would be equal to zero, then all iterations are equal to zero. In this way we obtain only trivial solution. That is way it has tried to find simple mean value formula, as in real integral. Mathematics Subject Classification: 34E05, 34K11 Keywords: The mean value theorem, Iteration, Sturm function, Complex differential equation, Oscillation
1
Introduction and Preliminary Notes
It is common that term mean value of complex integral has been introduced for closed contours (for the further details, see [2]), according to which f (z) is analytical or at least continuous, while inside contour can be discontinuous and even non-analitical. This was suggested by Cauchy’s fundamental theorem
J. Vujakovi´c
1818
f (z) dz = 0.
(1.1)
L (ς) is not Let z is point from the interior of contour L. Then function fς−z analytical all over inside L, so Cauchy’s fundamental integral formula is in effect
1 f (z) = 2πi
L
f (ς) dς . ς −z
(1.2)
If f (z) is analytical on L , then L can be replaced by the simplest closed contour, circle K of radius R with center at z. So standard formula(1.2) also applied for L ≡ K. Under mean value of function we consider value
K
f (ς) dς ς−z
divided with thelenght of circle arc S = 2Rπ. Having in (1.2) that ς = z + R eiϕ , 0 ≤ ϕ ≤ 2π, we obtain
1 f (z) = 2π
2π
f z + Reiϕ dϕ .
(1.3)
0
If f (z) = u (x, y) + iv (x, y) is analytical, then u (x, y) and v (x, y) are harmonic functions, so for them theorem on mean value also applies
1 u (z0 ) = 2π
2π
1 u z0 + reiϕ dϕ , v (z0 ) = 2π
0
2π
v z0 + reiϕ dϕ .
(1.4)
0
However, in solving ordinary differential equations in real area, formulas from (1.1) to (1.4) are of small importance because closed contour has not bee used. So, certain method for easier calculus or estimation of complex line integral has been searched for, and as well for real integral. For real integral, as is known, of great importance is first theorem on mean value of certain integral. That is why, for unclosed pathL, we need approximate value of integrals a (z) dz 2 ,
a (z) dz, L
L
L
za (z) dz 2 , whereat we shall examine L
L
approximation by module. Therefore, the following hypothesis applies.
The mean value theorem...
1819
Formula a (z) dz = a (ς) (z1 − z0 )
(1.5)
L
where 1. a (z) is analytical function in area G , where is placed curve L on which integration is conducted 2. z0 and z1 are initial and final limits of integral 3. ς is certain point from G, doesn’t have to be L, but which satisfies certain standards of approximation, first according to module and then according to argument. 4. A (z) is primitive function for a (z), which is possible to find by quadrature can be adopted as some kind of mean value of integral, if integration path is open line L. Fundaments for this hypothesis are: 1. If z1 = z0 , that is if L exists and closes down, then we obtain a (z) dz = 0, which is fundamental formula of Cauchy. L
x1 x0
2. For real z ≡ x we have section L = [x0 , x1 ] and formula (1.5) becomes a (x) dx = a (ς) (x1 − x0 ) , x0 ≤ ς ≤ x, and that is formula for mean value
of integral in real area. 3. If a (ς) is discontinuous in given area G where path L is placed, that (ς) , then Cauchy’s fundamental theorem or Cauchy’s theorem on is a (ς) = fς−z residues are valid. 4. Besides that, also are in effect formulas (1.3) and (1.4) on mean value of analytical function a (z), that is, on its harmonic parts u = Re a (ς) , v = Im a (ς). So, basic knowledge about analytics fit into assumption (1.5), but its construction demands existence of singular points from a (z) which don’t have to be on L. Thus, if L bends and closes and if the length of its arc is S = 2Rπ, then L becomes a circle K of length S, so in order to have first characteristic valid, ς must be out of circle K.
2
Main Results
These are the main results of the paper. Theorem 2.1. For integral of power function a (z) = z n , n ≥ 0 on arbitrary open curve, from z0 = 0 to point z1 , theorem on mean value is applied
J. Vujakovi´c
1820
z1 a (z) dz =
a (z) dz = a (ς) z1 ,
(2.1)
0
where
ς= √ n
z1 . n+1
(2.2)
Integral of each Jordan curve inside the circle |z| ≤ |z1 | = R can be replaced by integral on shortest path, that is, on direction Oz1 and (2.1), where the mid point on the path was ngiven with (2.2) and the mean value of the function is z1 z1 . = n+1 a (ς) = a √ n n+1 Proof. Let’s first observe the case n = 0, that is, let a a (z) = c is complex constant. Since the integral of analytic function does not depend on path L but only on primitive function A (z) = z for given a (z) = c, and from starting and final point, implies a (z) dz = cdz = c (z1 − z0 ). Here L
L
is c (z1 − z0 ) = a (ς) (z1 − z0 ), and that is, actually, formula (1.5) in which a (ς) = c and ς any point on segment z0 z1 . For n = 1, that is for function a (z) = z, implies a (z) dz = zdz = L
z 2 −z 2
L
0 (z12 − z02 ) = a (ς) (z1 − z0 ). Hence, since a (ς) = ς we have ς = 12 z11 −z00 = z1 +z . 2 Therefore, ς is arithmetic mean of final points z0 and z1 . Let’s further review the case n = 2. If a (z) = z 2 , then a (z) dz = L 2 z dz = 13 (z13 − z03 ), that is 13 (z13 − z03 ) = a (ς) (z1 − z0 ) or a (ς) = 13 (z12 + z0 z1
1 2
L
2
+z02 ) = ς 2 . From here, due to many-valued characteristics of squarer root we z 2 +z z +z 2 have two values for ς = ± 1 03 1 0 . Notice that ς 2 is arithmetic mean of three squares z12 , z0 z1 , z02 , taken in the starting and final point. We are Let a (z) = z n . Then a (ς) = with the procedure. continuing n z1 1 ς n = n+1 z1n−1 z0 + ... + z0n . For z0 = 0 implies ς = √ . Since z1n + n n+1 1 √ n n + 1 → 1 for n → ∞ the mean value ς → z1 , for n → ∞, on segment Oz1 , and a (ς) tends to a (z1 ). So, we obtained sequence {ς n } of points, z1 √ z1 √z1 , which on segment Oz1 tends to z1 . On the other hand, , z13 , √ 3 , ..., n 2 n+1 4 with induction it easy to prove that
The mean value theorem...
1821
z1
cdz = L
cdz = cz1 = a (ς) z1 =⇒ a (ς) = c = const 0 z1
zdz = L
zdz =
z12 z1 = a (ς) z1 = ςz1 =⇒ ς = 2 2
0
z1
z 2 dz = L
z 2 dz =
z13 z1 = a (ς) z1 = ς 2 z1 =⇒ ς = √ 3 3
z 3 dz =
z14 z1 = a (ς) z1 = ς 3 z1 =⇒ ς = √ 3 4 4
0
z1
z 3 dz = L
0
.. .
n
z1
z dz = L
z n dz =
z1n+1 z1 = a (ς) z1 = ς n z1 =⇒ ς = √ . n n+1 n+1
0
Let a (z) is arbitrary analytical function, defined by arbitrary convergent +∞ an z n . There is a question is it formula (2.1) valid or power series a (z) = n=0
some other variance of this formula, as well as where mid point ς is placed and what is the amount of mean value of a (ς). n Theorem 2.2. For arbitrary polynomial Pn (z) = ak z k applies fork=0
mula (2.1), for mean value of integral, where ς is certain mean value of module lesser than |z1 | = R1 , ( |ς| < R1 ), and which depends on z1 , that is on Pn (ak , z). Proof. For simple case, when P1 (z) = a0 + a1 z is linear function we have z1
P1 (z) dz = L
(a0 + a1 z) dz = a0 z1 + a1
z12 (2.1) = P1 (ς) z1 2
0
for P1 (ς) = a0 + a1 ς. From here ς = z21 . So mean value exists and is exactly in the middle of segment z0 z1 = Oz1 . Besides that, ς is inside the circle of radius |z1 | = R1 .
J. Vujakovi´c
1822 From the integral of squared function z1
P2 (z) dz = L
z2 z 3 (2.1) a0 + a1 z + a2 z 2 dz = a0 z1 + a1 1 + a2 1 = P2 (ς) z1 2 3
0
we obtain
a1 ς + a2 ς 2 = a1
z1 z2 + a2 1 . 2 3
(2.3)
Here implies equality
F (a1 , a2 , ς) = a1 ς + a2 ς 2 , G (a1 , a2 , z1 ) = a1
z1 z2 + a2 1 2 3
of two complex function with the same coefficients a1 , a2 , but with different complex arguments ς and z1 . On the basis of equality (2.3) we obtain F (a1 , a2 , ς) = G (a1 , a2 , z1 ), and here implies the equality of their modules |F (a1 , a2 , ς)| = |G (a1 , a2 , z1 )| .
(2.4)
In such way we can also estimate the maximum of modules, |F (a1 , a2 , ς)| ≤ |z 2 | |a1 | |ς| + |a2 | |ς 2 | and |G (a1 , a2 , z1 )| ≤ |a1 | |z21 | + |a2 | 31 . Here the limits are certain real positive numbers. Since the coefficients of limits |a1 | and |a2 | are mutual, and since we don’t know which of max |F (a1 , a2 , ς)| and max |G (a1 , a2 , z1 )| is larger limit, in order to maintain (2.5) it
take, sufficiently 2 z | | in special case, that |a1 | = |a2 |. Then |a1 | (|ς| + |ς 2 |) = |a1 | |z21 | + 31 , that is |ς| = |z21 | and |ς| = |z√13| . Since we have obtained the contradictory, formula (2.4) can be sustainable only if |ς| < |z1 |. Let’s check whether formula is applicable for cubic function. Integral of this function is
z1 P3 (z) dz =
L
0
(2.1) a0 + a1 z + a2 z 2 + a3 z 3 dz = P3 (ς) z1
The mean value theorem...
Here appears a0 z1 + a1
z12 2
1823
+ a2
z13 3
+ a3
z12
z14 4
= (a0 + a1 ς + a2 ς 2 + a3 ς 3 ) z1 , or after z3
cancellation with z1 = 0, a1 z21 + a2 3 + a3 41 = a1 ς + a2 ς 2 + a3 ς 3 . We again obtain two different complex functions with arguments z1 and ς, and with equal coefficients F (a1 , a2 , a3 , ς) = a1 ς + a2 ς 2 + a3 ς 3 , G (a1 , a2 , a3 , z1 ) = a1
z1 z2 z3 + a2 1 + a3 1 . 2 3 4
From F (a1 , a2 , a3 , ς) = G (a1 , a2 , a3 , z1 ) implies the equality of modules, and from comparison of coefficients, in order to have special case |a1 | = |a2 | = |a3 | 1| fulfilled, |ς| = |z21 | , |ς| = |z√13| , |ς| = |z√ 3 . Appears that, ς must be lesser than 4 |z1 | |z1 | , √3 2
and then |z√314| . Therefore |ς| < |z1 |. We are concluding that theorem applies for all forms of complex polynomial, because in general case if L is open arc of certain curve, we have Pn (z) dz = L
n L
k
ak z dz =
k=0
n k=0
ak
z k+1 (2.1) = Pn (ς) z1 . k+1
From here, after cancellation with z1 = 0 we obtain polynomial equation a0 + a1 ς + a2 ς 2 + · · · + an ς n − a0 − a1
z1 z2 z n+1 − a2 1 − · · · − an =0. 2 3 n+1
Let an = 0. Then according to fundamental theorem of Algebra, upper equation has exactly n different roots, real and complex. If we form in the same way functions F (ai , ς) = a1 ς + a2 ς 2 + · · · + an ς n , z1 z2 z n+1 , i = 1, n. G (ai , z1 ) = a1 + a2 1 + · · · + an 1 2 3 n+1 then from F (ai , ς) = G (ai , z1 ) follows |F (ai , ς)| = |G (ai , z1 )|. As a little while ago, we define limits max |F (a1 , . . . , an , ς)| = |a1 | |ς| + |a2 | |ς|2 + · · · + 2 n 1| |an | |ς|n and max |G (a1 , . . . , an , z1 )| = |a1 | |z21 | + |a2 | |z13| + · · · + |an | |zn+1 . From |F (ai , ς)| = |G (ai , z1 )| , i = 1, n, specially for |a1 | = |a2 | = . . . = |an |, appears |z1 | 1| = ... = √ , which is contradictory. So, none of above |ς| = |z21 | = |z√13| = |z√ n 3 n+1 4 mentioned equality is valid, but only |ς| < |z1 |.
J. Vujakovi´c
1824
3
Sturm functions
Now we can solve integrals in 2 2 w1 = 1 − a (z) dz + a (z) dz L
L
L
L n
w2 = z −
za (z) dz + L
L
L
L
L n
L
n-double integrals
(3.1)
a (z) dz
L
a (z) dz 2
a (z) dz ...
2
L
a (z) dz 2 · · · +
2
L
L
+ (−1)
and
za (z) dz 2 − · · · +
2 L
L
2
za (z) dz 2 (3.2)
a (z) dz ...
+ (−1)
L
L
L
L
n-double integrals
using theorem on mean value. We have the same situation as with iteration for real integrals, regardless of not having monotony
see [1],[3]) (for details (2.1) z2 a (z) dz 2 = a (z) dz dz = (a (ς 1 ) z) dz = a (ς 1 ) zdz = a (ς 1 ) 2!1 L L L L L L z4 a (z) dz 2 a (z) dz 2 = a (ς 1 ) a (ς 2 ) 4!1 L L L L z6 2 a (z) dz a (z) dz 2 a (z) dz 2 = a (ς 1 ) a (ς 2 ) a (ς 3 ) 6!1 L L
.. .
L L
L L
a (z) dz · · · 2
L
L
L
z 2n
1 a (z) dz 2 =a (ς 1 ) a (ς 2 ) . . . a (ς n ) (2n)!
L
n-double integrals
where all ς i , i = 1, n are placed in the same circle,|ς i | < |z1 | ≤ R1 , but can have different arguments. Besides that, when i → +∞ all ς i on their path tend to z1 , staying in the circle of radius R1 . Therefore, for (3.1) we obtain w1 = cosa(z) z = 1 − a (ς 1 )
z12 z4 z6 + a (ς 1 ) a (ς 2 ) 1 − a (ς 1 ) a (ς 2 ) a (ς 3 ) 1 + · · · 2! 4! 6!
The mean value theorem...
1825
or w1 =
+∞
k
(−1)
k
a (ς i )
i=1
k=0
z12k . (2k)!
(3.3)
We notice that in coefficients appear geometric mean in some mid points ς i , inside circle |z| < |z1 | = R1 . We shall mark these means in order with g0 = 1 g1 = a (ς 1 ) a (ς 1 ) a (ς 2 ) g2 = g3 = 3 a (ς 1 ) a (ς 2 ) a (ς 3 ) .. . gn = n a (ς 1 ) a (ς 2 ) . . . a (ς n ) . Even though the roots of complex number are multimorphic functions, they are in a certain way unambiguous, because all ς i , i = 1, n are inside the circle |z| < |z1 | = R1 , that is, if n → +∞ then series (3.3) gains more and n → z . Here appears that a (ς . . . a (ς n ) → more members, so ς n 1 1 ) a (ς 2 ) n n a (z a (z1 ). So 1 ) → a (z1 ) also for any branch of the function gn = n n a (ς k ) = lim gn = a (z1 ) applies. We have now that lim n→+∞
k=1
n→+∞
z2 z4 z6 (−1)k w1 = cosa(z) z = 1 − g1 1 + g22 1 − g33 1 + · · · = 2! 4! 6! k=0 +∞
Let’s also introduce geometric mean of these means g (ς) = such way, upper sum becomes
w1 = cosa(z) z ≈
+∞ k=0
(−1)k
2k g (ς)z1 (2k)!
= cos
√
2k gk z1 . (2k)!
√ n
g1 g2 . . . gn . In
g (ς)z1 ,
(3.4)
where symbol for approximately equal is valid because we have introduced one mean instead g1, g2 , . . . , gn . For solution w2 of equation w + a (z) w = 0, given with (3.2) with the same procedure we have similar estimations of multiple integrals
J. Vujakovi´c
1826
z3
za (z) dz 2 = a (ς 1 ) 3!1 L L z5 a (z) dz 2 za (z) dz 2 = a (ς 1 ) a (ς 2 ) 5!1 L L L L z7 2 a (z) dz a (z) dz 2 za (z) dz 2 = a (ς 1 ) a (ς 2 ) a (ς 3 ) 7!1 L L
.. .
L L
L L
a (z) dz · · · 2
L
L
L
z 2n+1
1 za (z) dz 2 = a (ς 1 ) a (ς 2 ) . . . a (ς n ) (2n+1)! .
L
n-double integrals
Here we also find second sine solution
w2 = sina(z) z =
+∞ k=0
(−1)
k
k
a (ς i )
i=1
z12k+1 . (2k + 1)!
(3.5)
If in the same way as before we introduce geometric means we obtain
w2 = sina(z) z ≈
1 g (ς)
+∞ k=0
(−1)k
2k+1 g (ς)z1 (2k + 1)!
1 sin = g (ς)z1 . g (ς)
Definition. Functions w1 and w2 given with (3.3) and (3.5) we call Sturm complex function with the basis a(z), which is analytical function.
4
Conclusion
The differences between Sturm real (for the details see [1]) and complex (for the details see [4])functions are as follows: 1. Real functions are unambiguous while complex are not. 2. In real a (x) is positive, in complex a (z) is entire function. 3. In the case of real, solutions are whole; while in complex sinusoidal solution can have omitted points. 4. Solutions are limited when x → ∞, and in complex do not have to be limited if z → ∞. 5. For a (x) > 0 oscillatory solutions are strictly separated from monotonous, when a (x) < 0, while in the case of complex, due to absence of relations “less” and “greater” , such division does not exist.
The mean value theorem...
1827
Contraction of operator for linear complex differential equation of second order w + a (z) w = 0, can easily be proven and the contraction coefficient can easily be found. Analogy with analytic sine and cosine in real sense has been shown in this paper. The problem remained to be shown is the oscillation of the solution, that is, locating of zeros. Acknowledgments. Financial support for this study was granted by the Ministry of Education, Science and Technology Development of the Republic of Serbia (Project Number TR 35030).
References [1] M. Leki´c, S. Cveji´c, D. Dimitrovski, Sturm theorems through iteration, Monography, University of Priˇstina, Faculty of Sciences and Mathematics, Kosovska Mitrovica, 2012. (in Serbian) [2] B. V. Shabat, Introduction to complex analysis, Providence, R. I: American Mathematical Society, 1992. [3] J. Vujakovi´c, M. Rajovi´c, D. Dimitrovski, Some new results on a linear equation of the second order, Computers and Mathematics with Applications 61 (2011), 1837-1843, doi: 10.1016/j.camwa.2011.02.012. [4] J. Vujakovi´c, Zeros solution of complex differential equations, Dissertation, University of Priˇstina, Faculty of Sciences and Mathematics, Kosovska Mitrovica, 2012. (in Serbian) Received: February 5, 2014
